30-60-90 과 45-45-90 — 한 번 외우면 즉시 풀이
[email protected], Geometry Calculator Developer & Online Math Educator 검수 마지막 업데이트 May 21, 2026
기하학, 삼각법, 물리학에서 두 직각삼각형이 너무 자주 등장해서 변의 비율을 외워두는 것이 좋습니다: 30-60-90 삼각형 (변 1 : √3 : 2)과 45-45-90 삼각형 (변 1 : 1 : √2)입니다. 각을 알아보면 계산 없이 변을 바로 알 수 있어 매번 피타고라스 계산을 절약할 수 있습니다.
| 이름 | 공식 | 비고 |
|---|---|---|
| 30-60-90 변의 비율 | 1 : √3 : 2 |
각각 30°, 60°, 90° 각의 대변입니다. 짧은 변, 긴 변, 빗변. |
| 30-60-90 긴 변 | long_leg = short_leg × √3 |
60°의 대변은 30°의 대변의 √3배입니다. |
| 30-60-90 빗변 | hypotenuse = 2 × short_leg |
빗변은 항상 가장 짧은 변(30°의 대변)의 정확히 두 배입니다. |
| 빗변으로부터 30-60-90 짧은 변 | short_leg = hypotenuse / 2 |
역: 빗변을 알면 그 절반이 30°의 대변을 제공합니다. |
| 빗변으로부터 30-60-90 긴 변 | long_leg = (hypotenuse / 2) × √3 |
짧은_변 = 빗변/2를 긴_변 = 짧은_변 × √3에 대입합니다. |
| 45-45-90 변의 비율 | 1 : 1 : √2 |
두 변이 같습니다(이등변 직각삼각형); 빗변은 변 × √2입니다. |
| 45-45-90 빗변 | hypotenuse = leg × √2 |
모든 이등변 직각삼각형에서 빗변은 각 변의 √2 ≈ 1.414배입니다. |
| 빗변으로부터 45-45-90 변 | leg = hypotenuse / √2 = hypotenuse·√2/2 |
역 — 빗변을 알면 √2로 나눕니다. |
| 정사각형으로부터 45-45-90 | half a square diagonally = 45-45-90 |
45-45-90 삼각형은 정사각형을 대각선으로 자른 정확히 절반입니다. 이것이 s² + s² = (s√2)² = 2s²인 이유입니다. |
| 정삼각형으로부터 30-60-90 | half an equilateral cut by altitude |
30-60-90 삼각형은 정삼각형을 높이로 자른 절반입니다. 여기서 비율 1 : √3 : 2가 나옵니다. |
| sin(30°), cos(60°) | = 1/2 |
30-60-90 삼각형에서 직접 읽습니다: 대변/빗변 = 짧은_변/2 = (1)/(2) (짧은_변 = 1일 때). |
| sin(60°), cos(30°) | = √3/2 |
1:√3:2 삼각형에서 대변/빗변 = 긴_변/2 = √3/2. |
| sin(45°), cos(45°) | = √2/2 ≈ 0.707 |
45-45-90에서 각의 대변과 인접변은 모두 변(둘 다 = 1)이고 빗변 = √2입니다. 따라서 sin = 1/√2 = √2/2. |
| tan(30°) | = 1/√3 = √3/3 |
대변/인접변 = 짧은_변/긴_변 = 1/√3. |
| tan(45°) | = 1 |
대변/인접변 = 변/변 = 1. |
| tan(60°) | = √3 |
대변/인접변 = 긴_변/짧은_변 = √3/1. |
Efficiently solving for unknown dimensions in special right triangles requires understanding the specific input parameters for each triangle type. Our calculator simplifies this process by allowing you to derive all missing sides and angles from a single known value. To begin, identify which type of triangle you are working with: either a 45-45-90 triangle or a 30-60-90 triangle. These shapes have fixed angle measures and consistent side ratios, which allows for precise calculations without needing complex trigonometric functions.
First, enter the known side length into the designated input field. This value can be either a leg or the hypotenuse, depending on the information provided in your problem. For a 45-45-90 triangle, the two legs are always equal in length. If you know the length of one leg, the other leg is identical. If you know the hypotenuse, you can find the leg length by dividing the hypotenuse by √2. Conversely, for a 30-60-90 triangle, the sides follow a ratio of 1 : √3 : 2. The shortest side is opposite the 30-degree angle, the longer leg is opposite the 60-degree angle, and the hypotenuse is twice the length of the shortest side.
After entering your value, select the corresponding triangle type from the dropdown menu. This ensures the calculator applies the correct geometric relationships. Finally, click the calculate button. The tool will instantly compute the remaining side lengths and confirm the angle measurements. While these calculators rely on fixed ratios, they are mathematically equivalent to using trigonometric ratios like sine, cosine, and tangent, but with significantly fewer steps. Additionally, you can verify your results manually using the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (a² + b² = c²). This step-by-step approach ensures accuracy and saves time when solving geometry problems involving these specific triangular forms.
Even with standardized shapes, errors frequently occur when applying geometric principles to 30-60-90 triangle and 45-45-90 triangle problems. One of the most prevalent issues involves confusing the roles of the legs and the hypotenuse. In a 45-45-90 triangle, students often mistakenly assume the hypotenuse is equal to the leg length multiplied by two. However, the correct relationship is that the hypotenuse is the leg length multiplied by √2. Similarly, in a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg, not the longer leg. Misidentifying which side corresponds to which part of the ratio leads to significant calculation errors.
Another common pitfall is misapplying side ratios across different triangle types. The ratio 1 : √3 : 2 is exclusive to the 30-60-90 configuration, while the 45-45-90 triangle follows the 1 : 1 : √2 pattern. Using the wrong ratio set invalidates the entire solution. It is crucial to recognize the angle measures first before selecting the appropriate mathematical model. Furthermore, many learners forget that the hypotenuse is always the longest side in any right triangle. This fundamental property serves as a quick check for validity; if a calculated hypotenuse is shorter than a leg, the result is physically impossible. Relying solely on memorized shortcuts without verifying against the Pythagorean theorem can also lead to mistakes. Understanding the underlying geometry rather than just the formula helps prevent these errors. By carefully distinguishing between the leg and the hypotenuse and respecting the unique properties of each triangle type, you can ensure accurate results in your geometric computations.